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Mirrors > Home > MPE Home > Th. List > grpideu | Structured version Visualization version GIF version |
Description: The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.) |
Ref | Expression |
---|---|
grpcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpcl.p | ⊢ + = (+g‘𝐺) |
grpinvex.p | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
grpideu | ⊢ (𝐺 ∈ Grp → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 17790 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | mndideu 17664 | . 2 ⊢ (𝐺 ∈ Mnd → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐺 ∈ Grp → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ∃!wreu 3119 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 0gc0g 16460 Mndcmnd 17654 Grpcgrp 17783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-nul 5015 ax-pow 5067 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-iota 6090 df-fv 6135 df-ov 6913 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 |
This theorem is referenced by: (None) |
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